# 10.7 Complementary events

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CHAPTER 10. PROBABILITY
S
B
A
Step 2 : Determine the intersection
From the above figure we notice that there are no elements in common in A and B. Therefore the events are mutually exclusive.
10.7
Complementary events
DEFINITION: Complementary set
The complement of a set, A, is a different set that contains all of the elements that are not in A. We write the complement of A as A0 , or sometimes
as “not (A)”.
For an experiment with sample space S and an event A we can derive some identities
for complementary events. Since every element in A is not in A0 , we know that complementary events are mutually exclusive.
A ∩ A0 = ∅
Since every element in the sample space is either in A or in A0 , the union of complemen-
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CHAPTER 10. PROBABILITY
10.7
tary events covers the sample space.
A ∪ A0 = S
From the previous two identities, we also know that the probabilities of complementary
events sum to 1.
P (A) + P (A0 ) = P (A ∪ A0 ) = P (S) = 1
Video: VMcdl at www.everythingmaths.co.za
Example 8: Reasoning with Venn diagrams
QUESTION
In a survey 70 people were questioned about which product they use: A or B or
both. The report of the survey shows that 25 people use product A, 35 people use
product B and 15 people use neither. Use a Venn diagram to work out how many
people
1. use product A only
2. use product B only
3. use both product A and product B
SOLUTION
Step 1 : Summarise the sizes of the sample space, the event sets, their union
and their intersection
• We are told that 70 people were questioned, so the size of
the sample space is n(S) = 70.
• We are told that 25 people use product A, so n(A) = 25.
• We are told that 35 people use product B, so n(B) = 35.
• We are told that 15 people use neither product. This means
that 70−15 = 55 people use at least one of the two products,
so n(A ∪ B) = 55.
Focus Area: Mathematics
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• We are not told how many people use both products, so we
have to work out the size of the intersection, A∩B, by using
the identity
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
n(A ∪ B)
n(A) n(B) n(A ∩ B)
=
+
−
n(S)
n(S)
n(S)
n(S)
55
25 35 n(A ∩ B)
=
+
−
70
70 70
70
∴ n(A ∩ B) = 25 + 35 − 55
=5
Step 2 : Determine whether the events are mutually exclusive
Since the intersection of the events, A ∩ B, is not empty, the events
are not mutually exclusive. This means that their circles should overlap in the Venn diagram.
Step 3 : Draw the Venn diagram and fill in the numbers
S
A
B
20
5
30
15
1. 20 people use product A only.
2. 30 people use product B only.
3. 5 people use both products.
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CHAPTER 10. PROBABILITY
10.7
Exercise 10 - 3
1. A box contains coloured blocks. The number of each colour is given in
the following table.
Colour
Number of blocks
Purple
Orange
White
Pink
24
32
41
19
A block is selected randomly. What is the probability that the block will
be:
(a) purple
(b) purple or white
(c) pink and orange
(d) not orange?
2. A small school has a class with children of various ages. The table gives
the number of pupils of each age in the class.
3 years old
4 years old
5 years old
Male
2
7
6
Female
6
5
4
If a pupil is selected at random what is the probability that the pupil will
be:
(a) a female
(b) a 4 year old male
(c) aged 3 or 4
(d) aged 3 and 4
(e) not 5
(f) either 3 or female?
3. Fiona has 85 labelled discs, which are numbered from 1 to 85. If a disc
is selected at random what is the probability that the disc number:
(a) ends with 5
(b) is a multiple of 3
(c) is a multiple of 6
(d) is number 65
(e) is not a multiple of 5
(f) is a multiple of 4 or 3
Focus Area: Mathematics
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CHAPTER 10. PROBABILITY
(g) is a multiple of 2 and 6
(h) is number 1?
More practice
(1.) 00in
(2.) 00ip
video solutions
or help at www.everythingmaths.co.za
(3.) 00iq
Chapter 10 | Summary
Summary presentation: VMdve at www.everythingmaths.co.za
• An experiment refers to an uncertain process.
• An outcome of an experiment is a single result of that experiment.
• The sample space of an experiment is the set of all possible outcomes of that experiment. The sample space is denoted with the symbol S and the size of the sample
space (the total number of possible outcomes) is denoted with n(S).
• An event is a specific set of outcomes of an experiment that you are interested in.
An event is denoted with the letter E and the number of outcomes in the event
with n(E).
• A probability is a real number between 0 and 1 that describes how likely it is that
an event will occur.
&middot; A probability of 0 means that an event will never occur.
&middot; A probability of 1 means that an event will always occur.
&middot; A probability of 0,5 means that an event will occur half the time, or 1 time out
of every 2.
• A probability can also be written as a percentage or as a fraction.
• The relative frequency of an event is defined as the number of times that the event
occurs during experimental trials, divided by the total number of trials conducted.
• The union of two sets is a new set that contains all of the elements that are in at
least one of the two sets. The union is written as A ∪ B.
• The intersection of two sets is a new set that contains all of the elements that are in
both sets. The intersection is written as A ∩ B.
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